46.2.13 problem 13

Internal problem ID [9550]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.3 SOLUTIONS ABOUT SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number : 13
Date solved : Tuesday, September 30, 2025 at 06:20:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (\frac {5}{3} x +x^{2}\right ) y^{\prime }-\frac {y}{3}&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 44

Order:=8; 
ode:=x^2*diff(diff(y(x),x),x)+(5/3*x+x^2)*diff(y(x),x)-1/3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{4}/{3}} \left (1-\frac {1}{7} x +\frac {1}{35} x^{2}-\frac {1}{195} x^{3}+\frac {1}{1248} x^{4}-\frac {1}{9120} x^{5}+\frac {1}{75240} x^{6}-\frac {1}{693000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_1 \left (1-3 x +\operatorname {O}\left (x^{8}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 72

ode=x^2*D[y[x],{x,2}]+(5/3*x+x^2)*D[y[x],x]-1/3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (-\frac {x^7}{693000}+\frac {x^6}{75240}-\frac {x^5}{9120}+\frac {x^4}{1248}-\frac {x^3}{195}+\frac {x^2}{35}-\frac {x}{7}+1\right )+\frac {c_2 (1-3 x)}{x} \]

✓ Sympy. Time used: 0.418 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (x**2 + 5*x/3)*Derivative(y(x), x) - y(x)/3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \sqrt [3]{x} \left (\frac {x^{6}}{75240} - \frac {x^{5}}{9120} + \frac {x^{4}}{1248} - \frac {x^{3}}{195} + \frac {x^{2}}{35} - \frac {x}{7} + 1\right ) + \frac {C_{1} \left (1 - 3 x\right )}{x} + O\left (x^{8}\right ) \]